A Multiple-Comparison-Systems Method for Distributed Stability Analysis of Large-Scale Nonlinear Systems

TL;DR

This paper introduces a scalable, distributed method using multiple comparison systems and sum-of-squares techniques for stability analysis of large-scale nonlinear networks, demonstrated on Van der Pol oscillators.

Contribution

It proposes a novel multiple comparison systems approach combined with sum-of-squares methods for distributed stability analysis of large nonlinear systems.

Findings

The method is scalable and distributed.

It effectively analyzes stability through subsystem communication.

Demonstrated on a network of Van der Pol oscillators.

Abstract

Lyapunov functions provide a tool to analyze the stability of nonlinear systems without extensively solving the dynamics. Recent advances in sum-of-squares methods have enabled the algorithmic computation of Lyapunov functions for polynomial systems. However, for general large-scale nonlinear networks it is yet very difficult, and often impossible, both computationally and analytically, to find Lyapunov functions. In such cases, a system decomposition coupled to a vector Lyapunov functions approach provides a feasible alternative by analyzing the stability of the nonlinear network through a reduced-order comparison system. However, finding such a comparison system is not trivial and often, for a nonlinear network, there does not exist a single comparison system. In this work, we propose a multiple comparison systems approach for the algorithmic stability analysis of nonlinear systems. Using sum-of-squares methods we design a scalable and distributed algorithm which enables the computation of comparison systems using only communications between the neighboring subsystems. We demonstrate the algorithm by applying it to an arbitrarily generated network of interacting Van der Pol oscillators.